a compact metric space is second countable


Let (X,d) be a compactPlanetmathPlanetmath metric space, and for each nβˆˆβ„€+ define π’œn={B⁒(x,1/n):x∈X}, where B⁒(x,1/n) denotes the open ball centered about x of http://planetmath.org/node/1296radius 1/n. Each such collectionMathworldPlanetmath is an open cover of the compact space X, so for each nβˆˆβ„€+ there exists a finite collection ℬnβŠ†π’œn that X. Put ℬ=⋃n=1βˆžβ„¬n. Being a countableMathworldPlanetmath union of finite setsMathworldPlanetmath, it follows that ℬ is countable; we assert that it forms a basis for the metric topology on X. The first property of a basis is satisfied trivially, as each set ℬn is an open cover of X. For the second property, let x,x1,x2∈X, n1,n2βˆˆβ„€+, and suppose x∈B⁒(x1,1/n1)∩B⁒(x2,1/n2). Because the sets B⁒(x1,1/n1) and B⁒(x2,1/n2) are open in the metric topology on X, their intersectionMathworldPlanetmath is also open, so there exists Ο΅>0 such that B⁒(x,Ο΅)βŠ†B⁒(x1,1/n1)∩B⁒(x2,1/n2). Select Nβˆˆβ„€+ such that 1/N<Ο΅. There must exist x3∈X such that x∈B⁒(x3,1/2⁒N) (since ℬ2⁒N is an open cover of X). To see that B⁒(x3,1/2⁒N)βŠ†B⁒(x1,1/n1)∩B⁒(x2,1/n2), let y∈B⁒(x3,1/2⁒N). Then we have

d⁒(x,y)≀d⁒(x,x3)+d⁒(x3,y)<12⁒N+12⁒N=1N<ϡ⁒, (1)

so that y∈B⁒(x,Ο΅), from which it follows that y∈B⁒(x1,1/n1)∩B⁒(x2,1/n2), hence that B⁒(x3,1/2⁒N)βŠ†B⁒(x1,1/n1)∩B⁒(x2,1/n2). Thus the countable collection ℬ forms a basis for a topologyMathworldPlanetmath on X; the verification that the topology by ℬ is in fact the metric topology follows by an to that used to verify the second property of a basis, and completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof that X is second countable. ∎

It is worth nothing that, because a countable union of countable sets is countable, it would have been sufficient to assume that (X,d) was a LindelΓΆf space.

Title a compact metric space is second countable
Canonical name ACompactMetricSpaceIsSecondCountable
Date of creation 2013-03-22 17:00:49
Last modified on 2013-03-22 17:00:49
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 17
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 54D70
Related topic MetricSpace
Related topic Compact
Related topic Lindelof
Related topic Ball
Related topic basisTopologicalSpace
Related topic Cover
Related topic BasisTopologicalSpace